I cannot find anything about the calculational rules when working with gaussian processes. So, how do i calculate the mean and covariance of the product (and division) and sum (and difference) of two gaussian processes and when is the result still a gaussian process?
The sum of two Gaussian processes will be Gaussian (this assumes joint Gaussian, which includes independence as a special case.) (expectations sum, if independent covariance functions will sum also.)
But the product will not be Gaussian, see Product of two gaussian processes and When is the distribution of product of two normal distributed variables near normal distribution?.